On the other hand, if you write a 'fic with a character who's canonically and virtually unstoppable, then the question changes from "What can they do?" to "What can't they do?" It depends on the character's personality, and how it can happen in the structure of the canon.

Take the guy in my icon: He's Locke, from a Seventies manga series called "Choujin Locke", which means "Superman/Superhuman Locke". He's immortal and has psychic powers out the kazoo: Shapeshifting, healing, immense telepathic and telekinetic prowess, and he can survive in space without a suit, among other things.

But while he can kick ass and take names, and has, he doesn't until his life or someone else's is in danger, and he holds back in fights because he genuinely wishes he was somewhere else. (In fact, Locke often spends his time living anonymously.) He's gotten almost fatally hurt several times, and even brainwashed on one occasion, despite the fact he could have avoided that.

So even though Locke is the Godmodder From Hell, there are "chinks in his armor", and his personality to consider. 'Fics involving him or characters like him would still require a lot of balance to portray well or correctly without irritating the reader. (And considering his canon stories lasted thirty years, it can be done! XD)